f(s+C,t+c) = g(t+E), so lim f(s+c,t+E) lim g(t+e) g(t) = f(s,t),
E+0 E40
so f is not right continuous. In the next section, we shall give
sufficient additional conditions on f for the converse to hold.
2.3 Functions of Two Variables with Finite Variation
As the example in the previous section (at the end) slows, the
requirement that VarR(f) < m on bounded rectangles R is by itself
insufficient to give all the properties necessary to associate a
Stieltjes measure to it. In order to deduce properties of f from its
variation, we need some extra conditions. It seems natural to require
that each of the one-dimensional paths also have finite variation, but
we do not need quite that much. In fact, if VarR(f) < m on all bounded
2
rectangles RC:R and if the one-dimensional path f(.,t ): R + E
has finite variation for some t0, then the paths f(',t) have finite
variation for all t. More precisely, for any s>0, we have
Var os]f(-,t) < Var s]f(.,t ) + Var ( ,t ) ( (f).
(Note: We replace the second term by Var[(, t)(,t f) if
t < 0t). To see this, let o: O0 ( sO < s < ... < s s sbe a
partition of [O,s] (Figure 2-8): We have, for each i, 0 S i1 n-1,
If(s i+ ,t)-f(si,t)l = If(si+ ,t)-f(s ,t)-f(s +1 ,t0)+f(si,tO)
+f(si+ ,t0)-f(sito)
SIf (s+ ,t)-fst)(s ,t)-f(si+,t)+f(si,to)
+ If(si+ ,t0)-f(si ,t )